Hi Thunderbird upgraded to version 45.0 and was no longer compatible with the previous version of BirdieSync. I upgraded BirdieSync to 3.0.1.1 as requested. However now I cannot get any connection as BirdieSync (in Thunderbird) displays the message, "Impossible to connect to server 127.0.0.1". Why is it trying to connect to the loopback address? Any advice on recovering from this situation?

If Thunderbird is located on the same PC as BirdieSync server, it tries to connect on the loopback address (127.0.0.1).

Can you confirm that BirdieSync server (where you configure BirdieSync synchronization) was running ? Thunderbird was not set in off-line mode (Thunderbird menu File/Offline/Work offline) and could download mails from the Internet ?
Do you have the same problem if you temporarily disable your firewall ?
Can you try to restart BirdieSync server (exit with menu "File/Exit") and Thunderbird ?
If you still have the problem, could you build a log archive with BirdieSync menu "Help/Archive logs..." and send BirdieSyncLog.zip to support(at)birdiesync.com ?

Thunderbird happily sending and receiving emails. Firewall happy, too. PC rebooted and Thunderbird restarted.
However, your request "Can you confirm that BirdieSync server (where you configure BirdieSync synchronization) was running ?" gave the clue. The process BirdieSync.exe was not running, so I started it and there was a connection made. Still some problems, which I will investigate and post here on further.
Why is it that the process is not started automatically? I never had to start the process manually before?

I've upgraded to 45.0. I've upgraded to BS 3.0.1.1. Now Thunderbird says: BirdieSync incompatible. It says I'm still using 3.0.0.9. I've tried re-installing BS, following all the instructions, but Thunderbird won't see the new version.

If BirdieSync add-on is not displayed among other add-ons in Thunderbird (menu Tools/Add-ons) or an incorrect version is displayed (for instance here 3.0.0.9 instead of 3.0.1.1), there might be a problem with Thunderbird add-on manager.
Here is what I suggest:

As usual, I suggest you to backup your data as mentioned in this FAQ topic.

Uninstall BirdieSync on the PC (not on the mobile device) and ensure that "Keep BirdieSync user data" is ticked when asked during uninstallation, to keep your current configuration.

Then launch Thunderbird to force Thunderbird add-on manager to take into account the removal of BirdieSync add-on (this step is important to force the removal of BirdieSync add-on: do not reinstall BirdieSync before doing it).

Then close Thunderbird and reinstall BirdieSync on the PC.
Then relaunch Thunderbird and check if BirdieSync add-on is now present in the add-on list, with the correct version (3.0.1.1 in your case).
You may have to enable it if it is still disabled.

Thank you. That's done it. The uninstall route did cross my mind, but I was nervous of getting it wrong. Having the proper instructions gave me all the confidence I needed...

I did think for a moment that this update had cured the connection problem I have, but it hasn't. For some reason, though my phone connects through either wifi circuit in my house, it only finds BirdieSync through the downstairs TP-Link, & can't sync through the upstairs Plusnet wifi box. I assume it's some firewall thing, though BirdieSync seems to be allowed. Most confusing.